Generalization of an Existence Theorem for Variational Inequalities

Z.H. Huang
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Z.H. Huang: Institute of Applied Mathematics

Journal of Optimization Theory and Applications, 2003, vol. 118, issue 3, No 5, 567-585

Abstract: Abstract By using the concept of exceptional family of elements, Zhao proposed a new existence theorem for variational inequalities over a general nonempty closed convex set (Ref. 1, Theorem 2.3), which is a generalization of the well-known Moré's existence theorem for nonlinear complementarity problems. The proof of Theorem 2.3 in Ref. 1 depends strongly on the condition 0∈K. Since this condition is rather strict for a general variational inequality, Zhao proposed an open question at the end of Ref. 1: Can the condition 0∈K in Theorem 2.3 be removed? In this paper, we answer this open question. Furthermore, we present the new notion of exceptional family of elements and establish a theorem of the alternative, by which we develop two new existence theorems for variational inequalities. Our results generalize the Zhao existence result.

Keywords: Variational inequalities; complementarity problems; exceptional family of elements; existence theorem (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1023/B:JOTA.0000004871.55273.15

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